For a balanced three-phase load, real power P is given by which expression?

For a balanced three-phase load, real power P is given by which expression?

• A. P = V_L × I_L × cosφ
• B. P = 3 × V_L × I_L × cosφ
• C. P = V_Ph × I_Ph × cosφ
• D. P = √3 × V_L × I_L × cosφ

Answer: (D)

Real power in a three-phase system comes from the power actually delivered by each phase, which is V_phase × I_phase × cosφ per phase. In a balanced load, all three phases contribute equally, so the total real power is three times that: P_total = 3 V_Ph I_Ph cosφ. To express this with line quantities, use the relationships between phase and line values. For a Y-connected system, V_Ph = V_L/√3 and I_Ph = I_L, giving P_total = 3 (V_L/√3 × I_L cosφ) = √3 V_L I_L cosφ. For a Δ-connected system, V_Ph = V_L and I_Ph = I_L/√3, yielding the same result: P_total = 3 (V_L × I_L/√3 cosφ) = √3 V_L I_L cosφ. Therefore, when using line voltage, line current, and the load power factor, the real power is P = √3 V_L I_L cosφ. This is why the expression with the √3 factor is the correct one for a balanced three-phase load.

Real power in a three-phase system comes from the power actually delivered by each phase, which is V_phase × I_phase × cosφ per phase. In a balanced load, all three phases contribute equally, so the total real power is three times that: P_total = 3 V_Ph I_Ph cosφ.

To express this with line quantities, use the relationships between phase and line values. For a Y-connected system, V_Ph = V_L/√3 and I_Ph = I_L, giving P_total = 3 (V_L/√3 × I_L cosφ) = √3 V_L I_L cosφ. For a Δ-connected system, V_Ph = V_L and I_Ph = I_L/√3, yielding the same result: P_total = 3 (V_L × I_L/√3 cosφ) = √3 V_L I_L cosφ. Therefore, when using line voltage, line current, and the load power factor, the real power is P = √3 V_L I_L cosφ. This is why the expression with the √3 factor is the correct one for a balanced three-phase load.